Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem

We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the Optimal Transport problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid stepsize vanishes and we show with numerical experiments that it is able to reproduce subtle properties of the Optimal Transport problem

Sequential products in effect categories

A new categorical framework is provided for dealing with multiple arguments in a programming language with effects, for example in a language with imperative features. Like related frameworks (Monads, Arrows, Freyd categories), we distinguish two kinds of functions. In addition, we also distinguish two kinds of equations. Then, we are able to define a kind of product, that generalizes the usual categorical product. This yields a powerful tool for deriving many results about languages with effects

Patterns for computational effects arising from a monad or a comonad

This paper presents equational-based logics for proving first order properties of programming languages involving effects. We propose two dual inference system patterns that can be instanciated with monads or comonads in order to be used for proving properties of different effects. The first pattern provides inference rules which can be interpreted in the Kleisli category of a monad and the coKleisli category of the associated comonad. In a dual way, the second pattern provides inference rules which can be interpreted in the coKleisli category of a comonad and the Kleisli category of the associated monad. The logics combine a 3-tier effect system for terms consisting of pure terms and two other kinds of effects called 'constructors/observers' and 'modifiers', and a 2-tier system for 'up-to-effects' and 'strong' equations. Each pattern provides generic rules for dealing with any monad (respectively comonad), and it can be extended with specific rules for each effect. The paper presents two use cases: a language with exceptions (using the standard monadic semantics), and a language with state (using the less standard comonadic semantics). Finally, we prove that the obtained inference system for states is Hilbert-Post complete

Breaking a monad-comonad symmetry between computational effects

Computational effects may often be interpreted in the Kleisli category of a monad or in the coKleisli category of a comonad. The duality between monads and comonads corresponds, in general, to a symmetry between construction and observation, for instance between raising an exception and looking up a state. Thanks to the properties of adjunction one may go one step further: the coKleisli-on-Kleisli category of a monad provides a kind of observation with respect to a given construction, while dually the Kleisli-on-coKleisli category of a comonad provides a kind of construction with respect to a given observation. In the previous examples this gives rise to catching an exception and updating a state. However, the interpretation of computational effects is usually based on a category which is not self-dual, like the category of sets. This leads to a breaking of the monad-comonad duality. For instance, in a distributive category the state effect has much better properties than the exception effect. This remark provides a novel point of view on the usual mechanism for handling exceptions. The aim of this paper is to build an equational semantics for handling exceptions based on the coKleisli-on-Kleisli category of the monad of exceptions. We focus on n-ary functions and conditionals. We propose a programmer's language for exceptions and we prove that it has the required behaviour with respect to n-ary functions and conditionals.Comment: arXiv admin note: substantial text overlap with arXiv:1310.060

A duality between exceptions and states

In this short note we study the semantics of two basic computational effects, exceptions and states, from a new point of view. In the handling of exceptions we dissociate the control from the elementary operation which recovers from the exception. In this way it becomes apparent that there is a duality, in the categorical sense, between exceptions and states

Decorated proofs for computational effects: Exceptions

We define a proof system for exceptions which is close to the syntax for exceptions, in the sense that the exceptions do not appear explicitly in the type of any expression. This proof system is sound with respect to the intended denotational semantics of exceptions. With this inference system we prove several properties of exceptions.Comment: 11 page

States and exceptions considered as dual effects

In this paper we consider the two major computational effects of states and exceptions, from the point of view of diagrammatic logics. We get a surprising result: there exists a symmetry between these two effects, based on the well-known categorical duality between products and coproducts. More precisely, the lookup and update operations for states are respectively dual to the throw and catch operations for exceptions. This symmetry is deeply hidden in the programming languages; in order to unveil it, we start from the monoidal equational logic and we add progressively the logical features which are necessary for dealing with either effect. This approach gives rise to a new point of view on states and exceptions, which bypasses the problems due to the non-algebraicity of handling exceptions

Adjunctions for exceptions

An algebraic method is used to study the semantics of exceptions in computer languages. The exceptions form a computational effect, in the sense that there is an apparent mismatch between the syntax of exceptions and their intended semantics. We solve this apparent contradiction by efining a logic for exceptions with a proof system which is close to their syntax and where their intended semantics can be seen as a model. This requires a robust framework for logics and their morphisms, which is provided by categorical tools relying on adjunctions, fractions and limit sketches.Comment: In this Version 2, minor improvements are made to Version

Porteurs de pays à l’air libre : jeu et enjeux des pièces asymétriques dans la musique traditionnelle du Québec

La version intégrale de cette thèse est disponible uniquement pour consultation individuelle à la Bibliothèque de musique de l’Université de Montréal (www.bib.umontreal.ca\MU)Le répertoire de musique traditionnelle instrumentale québécoise comporte de nombreux exemples de pièces asymétriques, familièrement appelées « tounes croches » par les musiciens. Il s’agit de mélodies qui ne respectent pas le modèle carré dominant de 16 ou 32 temps par section. Sans être unique au Québec, la performance de pièces asymétriques soulève de nombreuses questions ethnomusicologiques. Les principaux objectifs de cette recherche sont, d’une part, de comprendre le « dialecte » musical asymétrique et, d’autre part, d’examiner divers enjeux reliés au jeu de pièces asymétriques chez les musiciens traditionnels québécois actuels. Suite à une vue d’ensemble sur la musique traditionnelle québécoise couvrant l’histoire, le répertoire, les styles de jeu et les contextes de jeu, les pièces asymétriques sont analysées en tant qu’objet musical. Basée sur les enregistrements historiques du Gramophone virtuel, la mise en comparaison de versions symétriques et asymétriques de pièces sert à élaborer une typologie comprenant trois grandes catégories : syntaxiques, morphologiques et pulsatives. La vingtaine de types d’asymétries syntaxiques répertoriés impliquent l’allongement ou l’écourtement dans une section. Elles peuvent survenir à différentes positions de la section, bien que les asymétries finales soient les plus courantes. Les asymétries morphologiques concernent la macrostructure des pièces telle que l’ordre de jeu des sections et le nombre de reprises de chacune d’elles. Une analyse de l’asymétrie dans d’autres traditions musicales d’Europe et d’Amérique du Nord est aussi réalisée. Elle permet d’affirmer que nous avons affaire à un système musical cohérent et en partie distinct dans le cas québécois. Les enjeux originels, identitaires, esthétiques et de réalisation sont ensuite explorés en se basant sur les résultats d’une enquête menée auprès de 18 musiciens traditionnels. Ces derniers expliquent l’origine des pièces asymétriques par des raisons cognitives, chorégraphiques et artistiques. Leur propos servent à démontrer que le jeu de pièces asymétriques touche à de nombreuses dimensions identitaires : nation, région, génération, communauté contextuelle et individu. Un choix esthétique, un désir de liberté dans la création, un lien au passé et un besoin de distinction d’un corpus de musique traditionnelle standardisé motivent les musiciens traditionnels québécois d’aujourd’hui à perpétuer et à enrichir le répertoire de pièces asymétriques.There are numerous examples of asymmetrical tunes in Québécois traditional music repertoire. Familiarly called “tounes croches” (crooked tunes) by musicians, those melodies do not respect the dominant square model consisting of 16 or 32 beats per section of a tune. Without being exclusive to the Quebec tradition, the performance of crooked tunes raises several ethnomusicological questions. The main objectives of this research were to study the musical “dialect” found in asymmetrical tunes, and to look at various issues concerning the performance of asymmetrical tunes by contemporary Québécois traditional musicians. After a chapter that presents an overview of the history, repertoire, playing styles, and performance contexts of Québécois traditional music, asymmetrical tunes are analyzed as a musical object. The comparison of square and asymmetrical versions of tunes, taken mainly from the historical recordings available on the Virtual Gramophone, yields a typology that considers three main categories of asymmetries: syntactical; morphological; and pulsative. Syntactical asymmetries include about twenty different types, all involving a lengthening or a shortening at various positions of a section. Asymmetries at the end of a section are the most common. Morphological asymmetries concern the macrostructure of tunes such as the playing order of sections or the number of their repeats. Asymmetrical tunes in the repertoire of other musical traditions from Europe and North America are also analyzed. This allows us to confirm the existence of a coherent asymmetrical musical system that is partly distinct in the case of the Québécois tradition. Origin, identity, aesthetic, and performing issues are next studied based on the results of a survey done among 18 traditional musicians. Those surveyed explain the origin of asymmetrical tunes by cognitive, choreographic, and artistic reasons. Their discourse indicates that the performance of asymmetrical tunes touches several dimensions of their identity such as national, regional, generational, contextual and individual. An aesthetic choice, a desire for freedom in creation, a link to the past, and a need of distinction from a standardized traditional music repertoire motivate contemporary Québécois traditional musicians to perpetuate and to enrich the repertoire of asymmetrical tunes